Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order

In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x is an element of [0, L], x = x(x, y, z) and t is an element of [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space fractional derivatives of order alpha, for 0 < alpha < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method; and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < alpha < 1) and super-diffusive (1 < alpha < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed. (C) 2016 Elsevier B.V. All rights reserved.